Bethesda put a vast amount of effort into the NPC Serana. She is arguably the NPC with the most complex AI in the entire game. While other NPCs specialize in a specific behavior (merchants, mercenaries, cooks, miners, etc.), Serana masters all of these behaviors in addition to her quest-related actions. This becomes highly apparent when taking her along as a follower.

The term follower applies to Serana more literally than to any other NPC. She does not just walk behind you; she accompanies you. Unlike Housecarls or mercenaries who follow every step and execute every command precisely, Serana's behavior appears unpredictable and independent. If you tell Lydia to "Wait here," she stands exactly in that spot and waits. If you tell Serana to "Wait here," she will stay in the area, but she does not stand at attention; instead, she interacts with the environment. She walks around, leans against a wall, sits on the stairs, or sits cross-legged on the floor.

She also does this automatically if you stop anywhere for 5 seconds. When you want to move on, you often have to look around for her first - she might be chopping wood or mining ore somewhere (which she starts doing just 5 minutes after awakening). Inside buildings, she looks for a place to sit, even in adjacent rooms. In shops, she sometimes stands behind the counter and acts as a merchant. If she finds an Arcane Enchanter, she focuses on it or mixes something together at an alchemy lab. She does all of this on her own, without being prompted. In taverns, she sits at the bar and has a drink. Or she joins other guests, joins the celebration, dances, and applauds the bard. Along the way, she occasionally comments on the weather or the surroundings. She also behaves accordingly at both altars of Molag Bal - at the one in Markarth, she mimics the behavior of the priest Logrolf, while she kneels devoutly before the one in the Volkihar Cathedral. In High Hrothgar, she greets the Greybeards respectfully with a nod and sometimes even immerses herself in meditation.

You can give Serana alternative clothing, but the items must surpass her Royal Vampire Armor, which makes her appear quite picky.
She also possesses a unique set of gestures. However, in some dialogues, the arm-waving becomes so excessive that one might think she is about to wet her pants.

Pros:
As a combat follower, she is exceptionally valuable. She levels up with you up to level 50 and masters scaling spells such as Ice Spike, Lightning Bolt, and Vampiric Drain. She is also proficient with a blade and in sneaking. A crucial advantage of Serana is her immortality. Even after the Dawnguard quests, she cannot die - not even by the player's hand (unlike Lydia). She is one of the strongest companions in the game.
She can be persuaded to cure her vampirism. However, as long as she remains a vampire, she can transform the player into a Vampire Lord at any time upon request. Her pure vampire blood also plays a major role in the prophecy. Thus, her vampirism provides benefits, especially since she refuses to marry anyway.
Serana can act as a permanent second follower (or a third, if a dog is also present). This allows you, for example, to pack 3 inventories full of loot.

Cons:
Her unprompted interaction with environmental objects can lead to unnecessary interruptions in the flow of the game or even cause damage. For example, if you finish shopping and leave a store, Serana might not follow right away because she needs to finish her current action first (sitting, cooking, mixing potions, etc.). It can take a while before she leaves the shop and follows you. Even fast travel does not help in this case. She will eventually exit the shop and make her way to the player on foot.
Often, you wish she would just stand still. However, she rarely does and sometimes pushes the player. This is fatal at cliffs or in front of pressure plate traps. She frequently triggers traps herself, and not just due to her restless wandering. For instance, inside Yngol Barrow, you must solve a rotating pillar puzzle before flipping the switch. There is no chance with Serana. You will not be able to stop her from messing with the lever. The poisoned dart trap is constantly triggered.
In combat, she regularly raises the dead. This is actually unnecessary since she is powerful enough on her own. She casts this spell "instinctively" and as an area-of-effect spell. If an enemy appears somewhere on the horizon, undead foxes or some skeletons - which you otherwise would have barely noticed - are suddenly standing by her side. She even raises dead she cannot see, including behind walls or in inaccessible areas. Often, you want to loot a dead bandit but cannot because Serana just reanimated them. In narrow tunnels, these reanimated corpses become an unnecessary obstruction.
If Serana is present as a second follower, friendly fire can occur in battle, causing both followers to attack each other. Serana will eventually kill the other one.
Some Dawnguard quests are designed so that Serana "forcibly accompanies" the player. The "Wait here" dialogue option is not available during those times. However, just like with other followers, you can put her into command mode and point to a spot where she should wait. She will comply, but unfortunately, there is no "Come with me" dialogue option in these quests (and a quest completion is only possible with an active Serana). This wait mode can be ended by waiting yourself for three days. There are, however, special quest situations that automatically end the wait mode, such as retrieving Auriel's Bow from the shrine.

As good as Serana is in combat and as refreshing as her restlessness can be - over time, the disadvantages can become increasingly annoying.